The monotony and continuity of functions
A sufficient condition for increasing functions
If at each point of the interval 
, the function is increasing on this interval
A sufficient condition for decreasing functions
If at each point of the interval 
, the function is decreasing on this interval
Remark. These conditions are only sufficient but not necessary conditions for growth and decreasing functions
A necessary and sufficient condition of constancy of the function
A function 
is constant on an interval 
if and only if when 
all points of the interval
The extrema (maximums and minimums) of the function
The maximum point
Definition: the Point 
of defining functions is called a maximum point of this function if there 
is a neighborhood 
of the point 
that for all 
from this neighborhood inequality 
— the maximum point
— max
The point of a minimum
Definition: the Point of defining functions is called the minimum point of this function if there is a neighborhood of the point that for all from this neighborhood inequality 
— minimum point
— minimum
The critical point
Definition: the Interior points of the domain of definition of functions where the derivative is zero or does not exist are called critical
Necessary condition for an extremum
— the extremum point 
or 
not exists
(but not at each point where or not there will be an extremum!)
Sufficient condition of extremum
at the point of sign 
changes 
at 
point of maximum
at the point the sign changes 
at the 
point of minimum
An example of a graph of a function 
that has an extremum
that has an extremum
— critical point
The study of the function on the monotonicity and extrema
- Find the domain of definition and the intervals on which the function is continuous
- To find the derivative
- Find the critical points, i.e. the interior points determining where
or not there - Denote the critical point on the domain of definition, find sign of derivative and nature of function at each interval, which splits the definition area
- For each critical point determine whether it is high or low or is not an extremum point
- Record porni the result of the study (intervals of monotonicity and extrema)
Example. 
Scope: 
The function is continuous at every point of its domain of definition
or not there
there is in the entire scope
when 
increases with 
and 
subsides when 
Extreme points: 
Extremes: 
Maximum and minimum values of continuous functions on the interval
Property: If the function is continuous on an interval and has therein a finite number of critical points, then it attains its maximum and minimum values on this interval either at a critical point belonging to this interval or at the endpoints of the interval
Finding maximum and minimum values of continuous functions on the interval
- To find the derivative
- Find the critical points (
or not exists) - Choose critical points that belong to a given segment
- To calculate the function values at critical points and the endpoints of the interval
- Compare the two values and choose the smallest and largest
Example. 
when 
if 
and when 
A given segment 
belongs to only critical point
